by Mike Ossipoff

To write to Mike write nkklrp before the "@" sign, and
then write hotmail.com after the "@" sign. Or write
ossipoff2002 before the "@" sign, and then write
yahoo.com after the "@" sign.

Here's an example for the purpose of demonstrating
the application of PC, RP, SD, SSD, CSSD, &
BeatpathWinner to a set of rankings.

In this example, SSD, CSSD, and BeatpathWinner choose
one winner; PC & SD choose a different winner; and RP
returns a tie between those 2 winners.

If you're just interested in the examples themselves,
there's no reason to not skip from here directly
to the heading "Now the example".

As was mentioned at the Condorcet page, though
RP, SD, SSD, CSSD, & BeatpathWinner choose from the
Schwartz set in public elections, because there are
no pairwise ties when there are so many voters,
it's possible, in small committee voting, for
RP & SD to choose outsidet the Schwartz set. This
example was chosen to demonstrate that.

The Schwartz set was defined in the definition of
SSD (Schwartz Sequential Dropping), at the Condorcet
page.

Incidentally, even in small committee elections,
all of the above-listed methods except for PC
always choose from the Smith set. The Smith set is
the smallest set of candidates such that every
candidate in the set pairwise-beats every candidate
outside of the set. X pairwise-beats Y if more
voters rank X over Y than Y over X.

In public elections, where there are no pairwise-ties,
the Smith & Schwartz sets are identical.

PC's ability to choose outside the Smith set, even
in public elections, can lead to embarrassing, though
not really important, transgression examples, which
can
be used against PC in a campaign for its adoption.
On the other hand, PC is by far the simplest Condorcet
version, which could be important in a campaign.

AB2 means that A beats B, with the defeat strength
of 2, because 2 people have ranked A over B (and
fewer have ranked B over A).

I don't yet have rankings for these pairwise defeats,
but it's a sure thing that this
example can happen, because it's been shown that,
for any set of pairwise preference vote totals,
there is some constant that we could add to them all
that would give a set of pairwise vote totals that's
consistent with some set of rankings. Since the order
of the defeats is what counts in this example,
of course the example would work just as well if
some constant were added to all of the pairwise
vote totals.

Of course if A beats B, I only list the number of
people ranking A over B. Of course some smaller number
ranked B over A, but it isn't necessary to specify
that number. That's because, with these methods,
we measure A's defeat of B by how many voters ranked
A over B.

The best way to look at examples like this is to
make a diagram in which the candidates are arranged
in small circular arrangements. Since this example
has two cycles, ABC & DEF, arrange those candidates
in 2 triangular sets on the page. Then draw arrows
to represent the defeats, so that if A beats B,
there's an arrow from A to B. Then label the arrows
with the defeat magnitudes.

In this example there are 2 cycles: the ABC cycle
and the DEF cycle. Every candidate in {A,B,C} pairwise
beats D & E, but ties F. It isn't necessary to specify
those
inter-cycle defeat magnitudes, because none of them
are in a cycle, and so they'll all be kept in RP,
and none will be dropped in SD or SSD or SSD, and
none have a return beatpath in BeatpathWinner.

Here are the defeats in the 2 cycles:

AB5, BC6, CA4 DE3, EF1, FD2

What do PC, SSD, CSSD, BeatpathWinner, RP, & SD do
in this example?

Again, if you're just interested in the examples
themselves, skip directly to where, a few paragaraphs
down, it says "So let's apply all 6 methods, in turn,
to the example:"

First of all, since nothing in {A,B,C} is beaten by
anything outside the set, {A,B,C} is an unbeaten
set. Since it doesn't contain a smaller unbeaten
set (every subset of {A,B,C} is beaten by someone
in {A,B,C}), {A,B,C} is an innermost unbeaten set.
It's the only one, and so it's the Schwartz set.

So we know that SSD, CSSD, & BeatpathWinner will
choose from {A,B,C}.

RP and SD can choose outside the Schwartz set, and so
they might choose outside {A,B,C}. They do in this
example.

Since F pair-ties A, B, & C, and is in a cycle with
D & E, all 6 candidates are the Smith set, because
no subset of the candidates pairwise-beat everyone
else.

EF1 is the weakest defeat. When it's dropped, F
has no defeats, and therefore wins. We haven't
specified the inter-cycle defeats, but their
magnitudes
don't matter, since all the candidates have bigger
defeats than F does.

SSD:

The smallest defeat among the Schwartz set is CA4.
When we drop it, A is unbeaten, and so A wins.

CSSD:

When we drop CA4, the Schwartz set now is {A}.
Obviously there are now no defeats in the Schwartz
set, and so its member, A, wins.

BeatpathWinner:

BeatpathWinner is equivalent to CSSD, and so we know
it will pick A too. But that can also be shown from
BeatpathWinner's own rule: A beatpath's strength is
the strength of its weakest defeat. Following the
beatpaths from A to B, and from A to C, and from B to
A, and from C to A, it turns out that A has
stronger beatpaths to B & C than they have to A.

Likewise, A has beatpaths to D,E, & F, but they
have no beatpath to A. So, by BeatpathWinner's rule,
A wins, because no one has a beatpath win against A.
That can't be said for any other candidate.

SD:

Drop the weakest defeat that's in a cycle. That's
EF1. Now F is undefeated, and F wins.

RP:

RP insists on solving all cycles, because any
defeat that's contradicted by kept stronger defeats
should be considered nullified by them.

First of all, all the defeats from the ABC cycle to
the DEF cycle eventually get kept, since none of them
are in cycles. Of course none of those intercycle
defeats is to F.

So, just looking at the defeats in the cycles, we
start by considering the strongest one, BC6. It
doesn't cycle with kept defeats because there are
none yet, since we haven't yet considered any other
defeats. So we keep BC6.

Next we consider AB5. It too doesn't cycle with kept
defeats, since we've only kept one defeat, and you
can't make a cycle with fewer than 3 defeats. So
we keep AB5. (In general, when applying RP's defining
rule, we always keep the 2 strongest defeats for the
reasons given above).

Considering CA4, it cycles with AB5 & BC6, which have
been kept, and so we don't keep CA4. Since that's
A's only defeat, that means that A will be a winner.

Next we similarly keep DE3 & FD2, but not EF1, because
it would cycle with the 2 previously-kept defeats.

So when we've considered all the defeats, we have
2 candidates with no kept defeats: A & F.

It's a tie, and so we'd use a tiebreaker. I'd suggest,
in organizations or small committees, Random Ballot.
Whichever of {A,F} is ranked higher on a randomly-
chosen ballot wins.

RP's choice makes a lot of sense, because any
defeat that's constradicted by stronger defeats
is understandably considered to be nullified by them,
unless they themselves are nullified in that way.

But the better decisiveness of the other methods
counts in their favor, for practical purposes, in
a small committee vote. In a public election, of
course, there's no decisiveness difference.

And it can also be reasonably argued that the winner
should come from the initial Schwartz set. So RP's
appeal, and the criterion of choosing from the initial
Schwartz set can't be had in one method.

But we can define a method that says to use RP to
choose from the initial Schwartz set. In other words,
delete all the candidates who aren't in the initial
Schwartz set, and then apply RP to the undeleted
candidates. That would be my favorite for small
committees, ideally, but, with RP being somewhat
less decisive than the others, and subject sometimes
to the need to solve ties during the middle of the
count, when 2 defeats are equal. With the added
wordiness of dealing with these things, RP doesn't
look as practical for small committees as the other
methods do.
SD has much simplicity appeal in small committees
and organizations, though, if we disregarded
simplicity, I'd prefer the other methods to it.
But simplicity can be important when proposing a
voting system to an organization or a small committee,
or to the public.

But, for public elections, RP
is as briefly defined as SD, and RP is better.
For a public proposal, it's a choice between RP's
merit, and PC's especially extreme simplicity.